Adaptive random compiler for Hamiltonian simulation

TL;DR

Fixed-norm sampling in randomized Hamiltonian compilation can be suboptimal for continuous-variable and unbounded operators. The authors introduce Adaptive Random Compiler (ARC), which updates Hamiltonian-term sampling probabilities using low-order moment measurements to emphasize uncertain terms and extend to CV and hybrid systems. They derive an optimal distribution p_j ∝ sqrt(||D_jj(ρ)||) with a practical moment-based realization for pure states, and provide a complexity analysis showing tighter state-dependent circuit-depth bounds. Numerical simulations across discrete-variable, continuous-variable, and hybrid-variable models demonstrate substantial fidelity gains, highlighting ARC's potential to reduce circuit depth and expand the reach of randomized compilation on NISQ devices.

Abstract

Randomized compilation protocols have recently attracted attention as alternatives to traditional deterministic Trotter-Suzuki methods, potentially reducing circuit depth and resource overhead. These protocols determine gate application probabilities based on the strengths of Hamiltonian terms, as measured by the trace norm. However, relying solely on the trace norm to define sampling distributions may not be optimal, especially for continuous-variable and hybrid-variable systems involving unbounded operators, where quantifying Hamiltonian strengths is challenging. In this work, we propose an adaptive randomized compilation algorithm that dynamically updates sampling weights via low-order moment measurements of Hamiltonian terms, assigning higher probabilities to terms with greater uncertainty. This approach improves accuracy without significantly increasing gate counts and extends randomized compilation to continuous-variable and hybrid-variable systems by addressing the difficulties in characterizing the strengths of unbounded Hamiltonian terms. Numerical simulations demonstrate the effectiveness of our method.

Figures (5)

• Figure 1: Pseudocode for adaptive random compiler protocol.
• Figure 2: Fidelity comparison of the mixed-field Ising model Hamiltonian simulation. (a) Fidelity versus the number of evolution steps at a fixed step size $t/N=0.02$. (b) Fidelity as a function of step size with the total evolution time fixed at $t=1$. In both subfigures, the red solid line corresponds to the adaptive random compiler, while the green dot-dashed line shows results for the original random compiler protocol. The dotted line in (b) represents the fidelity extrapolated to zero step size. Other parameters are $L=4$, $J=1$, $h_x=0.5$, $h_z=0.3$, with initial state $\ket{0011}$.
• Figure 3: Driven Kerr oscillator Hamiltonian fidelity results. (a) Fidelity versus the number of evolution steps, with fixed step size $t/N=0.02$. (b) Fidelity dependence on step size for a fixed total evolution time $t=1$. The red solid line denotes our method, the green dot-dashed line indicates the original protocol with hard truncation, and the purple dashed line corresponds to the random compiler assigning equal weights to all Hamiltonian terms. The dotted line in (b) shows extrapolated fidelity at zero step size. The parameters are set as $\Delta=0.3$, $K=1$, $\epsilon=0.5$, initial state $(\ket{1}+\ket{5})/\sqrt{2}$, and Fock space truncation dimension $D=50$.
• Figure 4: Quantum Rabi model Hamiltonian fidelity comparison. (a) Fidelity as a function of the number of evolution steps with fixed step size $t/N=0.02$. (b) Fidelity versus step size at total evolution time $t=1$. In both subfigures, the red solid line shows our method results, the green dot-dashed line corresponds to the original hard truncation protocol, and the purple dashed line depicts the random compiler with equal weighting. The dotted line in (b) represents the zero step size extrapolation. Parameters used are $\omega=1$, $\Omega=1$, $g=0.2$, initial state $(\ket{2,0}+\ket{5,0})/\sqrt{2}$, and truncation dimension $D=50$.
• Figure 5: Dynamic adjustment of the probability distribution during the simulation of the Rabi model Hamiltonian using the adaptive random compiler. Subfigures (a) and (b) correspond to coupling strengths $g=0.2$ and $g=0.8$, respectively, while other parameters are fixed as $D=50$ (the Fock space truncation dimension), $\omega=1$, $\Omega=1$ and $t/N=0.02$. The initial state in both cases is $(\ket{2,0}+\ket{5,0})/\sqrt{2}$. The red solid line, green dot-dash line and purple dashed line represent the sampling probabilities $p_1$, $p_2$ and $p_3$ corresponding to the three Hamiltonian terms, respectively.